[Paper Review] Semiorthogonal decompositions in algebraic geometry

Alexander Kuznetsov|arXiv (Cornell University)|Apr 11, 2014

Algebraic structures and combinatorial models41 references104 citations

TL;DR

This paper reviews semiorthogonal decompositions in algebraic geometry, focusing on their construction via homological projective duality and categorical resolutions of singularities. It establishes that derived categories of certain Fano fourfolds, such as Pfaffian cubic fourfolds and degree 10 fourfolds in Gr(2,5), contain noncommutative K3 categories, with associated hyperkähler varieties arising from Fano schemes and double EPW sextics.

ABSTRACT

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.

Motivation & Objective

• To systematize and review known constructions of semiorthogonal decompositions in derived categories of coherent sheaves on algebraic varieties.
• To explore the role of homological projective duality as a powerful method for constructing such decompositions.
• To investigate the interplay between categorical resolutions of singularities and homological projective duality.
• To identify and analyze cases where semiorthogonal components are noncommutative K3 categories, particularly in Fano fourfolds.
• To explore geometric realizations of hyperkähler varieties from derived categories of Fano fourfolds via Fano schemes and double EPW sextics.

Proposed method

• Utilizes triangulated categories and derived categories of coherent sheaves on smooth projective varieties over a field k.
• Applies the concept of admissible subcategories and their left/right orthogonals to construct semiorthogonal decompositions.
• Employs exceptional collections and Lefschetz decompositions, particularly in the context of Grassmannians, quadrics, and projective bundles.
• Applies homological projective duality (HPD) to relate derived categories of dual varieties, especially in the case of Pfaffian cubic fourfolds and degree 10 fourfolds.
• Uses the base change formula for semiorthogonal decompositions under base change and derived pullback.
• Applies the theory of categorical resolutions of singularities to relate singular and smooth derived categories.

Experimental results

Research questions

• RQ1 How can semiorthogonal decompositions be systematically constructed for derived categories of algebraic varieties?
• RQ2 What is the role of homological projective duality in generating semiorthogonal decompositions, especially for Fano varieties?
• RQ3 In which cases do semiorthogonal components of derived categories of Fano fourfolds yield noncommutative K3 categories?
• RQ4 Can hyperkähler varieties be geometrically realized from derived categories of Fano fourfolds via Fano schemes or double covers?
• RQ5 What is the structure of the derived category of a 5-fold in Gr(3,7) with a rectangular Lefschetz decomposition, and does its hyperplane section yield a K3-type category?

Key findings

• The derived category of a Pfaffian cubic fourfold admits a semiorthogonal decomposition with a component equivalent to the derived category of a K3 surface, suggesting a noncommutative K3 structure.
• For the 4-fold of degree 10 in Gr(2,5), the derived category decomposes as Db(coh(Y)) = ⟨AY, OY, U∨Y, OY(1), U∨Y(1)⟩, where AY is a Calabi–Yau category of dimension 2.
• The category AY for the degree 10 fourfold is conjectured to be equivalent to the derived category of a K3 surface, and the Fano scheme of conics on Y fibers over a double EPW sextic, a hyperkähler fourfold.
• The 5-fold X ⊂Gr(3,7) has a conjectural rectangular Lefschetz decomposition Db(coh(X)) = ⟨B, B(1)⟩ with B generated by six exceptional objects, implying its hyperplane section Y has a semiorthogonal decomposition with a K3-type component AY.
• The Fano scheme of lines on a cubic fourfold is realized as a hyperkähler variety, and the double EPW sextic construction generalizes this to other fourfolds with similar Hodge diamonds.
• For a smooth projective variety X of index m with a rectangular Lefschetz decomposition, the derived category of a hyperplane section Yd has a semiorthogonal decomposition involving a Calabi–Yau component AYd when d divides m.